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When a portfolio includes options, the linear model is an approximation. It does not take account of the gamma of the portfolio. As discussed in Unit 14, delta is defined as the rate of change of the portfolio value with respect to an underlying market variable and gamma is defined as the rate of change of the delta with respect to the market variable. Gamma measures the curvature of the relationship between the portfolio value and an underlying market variable.

Figure 3 shows the impact of a nonzero gamma on the probability distribution of the value of the portfolio. When gamma is positive, the probability distribution tends to be positively skewed; when gamma is negative, it tends to be negatively skewed. Figures 16.4 and 16.5 illustrate the reason for this result. Figure 16.4 shows the relationship between the value of a long call option and the price of the underlying asset. A long call is an example of an option position with positive gamma. The figure shows that, when the probability distribution for the price of the underlying asset at the end one day is normal, the probability distribution for the option price is positively skewed. Figure shows the relationship between the value of a short call position and the price of the underlying asset. A short call position has negative gamma. In this case we see that a normal distribution for the price of the underlying asset at the end of one day gets mapped into a negatively skewed distribution for the value of the option position. The VaR for a portfolio is critically dependent on the left tail of the probability distribution of the portfolio value. For example, when the confidence level used is $99 \%$, the VaR is calculated from the value in the left tail below which there is only $1 \%$ of the distribution. As indicated in Figures .3 and 4 , a positive-gamma portfolio tends to have a less heavy left tail than the normal distribution. If we assume the distribution is normal, we will tend to calculate a VaR that is too high. Similarly, as indicated in Figures $16.3 \mathrm{~b}$ and 16.5, a negative-gamma portfolio tends to have a heavier left tail than the normal distribution. If we assume the distribution is normal, we will tend to calculate a VaR that is too low.For a more accurate estimate of VaR than that given by the linear model, we can use both deltail and gamma measures to relate SP to the Sxt’s. Consider a portfolio dependent on a single assetwhose price is S. Suppose that the delta of a portfolio is A and its gamma is T. From Appendix, the equation
$$8 \mathrm{P}=\mathrm{ASS}+\backslash \mathrm{r}(\mathrm{SS}) 2$$

## 金融代写|金融衍生品代写Financial Derivatives代考|MONTE CARLO SIMULATION

As an alternative to the approaches described so far, we can implement the modelbuildingapproach using Monte Carlo simulation to generate the probability distribution for SP. Supposewe wish to calculate a one-day VaR for a portfolio. The procedure is as follows:

1. Value the portfolio today in the usual way using the current values of market variables.
2. Sample once from the multivariate normal probability distribution of the \&,’s.10
3. Use the values of the Sxt’s that are sampled to determine the value of each market variable atthe end of one day.
4. Revalue the portfolio at the end of the day in the usual way.
5. Subtract the value calculated in step 1 from the value in step 4 to determine a sample SP.
6. Repeat steps 2 to 5 many times to build up a probability distribution for SP.

The VaR is calculated as the appropriate percentile of the probability distribution of SP. Suppose, For example, that we calculate 5,000 different sample values of SP in the way just described. The1-day $99 \% \mathrm{VaR}$ is the value of SP for the 50 th worst outcome; the 1-day VaR $95 \%$ is the value of SPfor the 250th worst outcome; and so on.11 The TV-day $\mathrm{VaR}$ is usually assumed to be the 1-day VaRmultiplied by V/V.12

The drawback of Monte Carlo simulation is that it tends to be slow because a company’scomplete portfolio (which might consist of hundreds of thousands of different instruments) has tobe revalued many times.13 One way of speeding things up is to assume that equation (8)describes the relationship between SP and the \&,’s. We can then jump straight from step 2 tostep 5 in the Monte Carlo simulation and avoid the need for a complete revaluation of theportfolio. This is sometimes referred to as the partial simulation approach

## 金融衍生品代写

$8 \mathrm{P}=\mathrm{ASS}+\backslash \mathrm{r}(\mathrm{SS}) 2$

## 金融代写|金融衍生品代写FINANCIAL DERIVATIVES代 考|MONTE CARLO SIMULATION

1. 使用市场变量的当前值以通常的方式对今天的投资组合进行估值。
2. 从 |\&,’s. 10 的多元正态概率分布中抽样一次
3. 使用采样的 Sxt 的值来确定一天结束时每个市场变量的值。
4. 以通常的方式在一天结束时重新评估投资组合。
5. 多次重复步骤 2 到 5 以建立 SP 的概率分布。
$V a R$ 计算为 SP 概率分布的适当百分位数。假设，例如，我们按照刚才描述的方式计算了 5,000 个不同的 SP样本值。第一天 $99 \% V a R$ 是第 50 个最 差结果的 SP 值； 1 天 VaR95\%是第 250 个最差结果的 SP 值；等等。11 电视日VaR通常假定为 1 天VaR 乘以 V/V.12
蒙特卡洛模拟的缺点是它往往很慢，因为公司的完整投资组合whichmightconsistofhundredsofthousandsofdifferentinstruments必须 多次重估。 13 加快速度的一种方法是假设等式8描述了 SP和 $\backslash \&$, 之间的关系。然后我们可以在蒙特卡洛模拟中从第 2 步直接跳到第 5 步，避免对投 资组合进行全面重估的需要。这有时被称为部分模拟方法

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。